子流形平均曲率向量场的线性相关性

刘进

数学学报 ›› 2013, Vol. 56 ›› Issue (5) : 669-686.

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PDF(598 KB)
数学学报 ›› 2013, Vol. 56 ›› Issue (5) : 669-686. DOI: 10.12386/A2013sxxb0067
论文

子流形平均曲率向量场的线性相关性

    刘进1,2
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Linear Dependence of Mean Curvature Vector Fields of Submanifold

    Jin LIU1,2
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摘要

在空间形式中,构造了子流形的一类泛函, 其包含 r极小泛函与体积泛函(极小)作为特殊情形, 此类泛函的临界点称之为-平行子流形. 对于-平行子流形, 给出了代数,微分和变分刻画. 更进一步, 研究了-平行子流形的稳定性, 证明了Simons型不存在定理: 在一定条件下-函数为正), 球面中不存在稳定的 -平行子流形.

Abstract

A class of functionals is found in space forms such that its critical points include r-minimal submanifolds and minimal submanifolds as special cases. The critical points are defined as -parallel submanifolds. We obtain algebraic, differential and variational characterizations of the -parallel submanifolds. Moreover, we prove a Simons' type nonexistence theorem which says that in the unit sphere there exists no stable -parallel submanifold with its corresponding -function positive.

关键词

(r+1,λ)-平行子流形 / (r,λ)-函数 / Qr 算子

Key words

(r+1,λ)-parallel submanifold / (r,λ)-function / Qr operators

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刘进. 子流形平均曲率向量场的线性相关性. 数学学报, 2013, 56(5): 669-686 https://doi.org/10.12386/A2013sxxb0067
Jin LIU. Linear Dependence of Mean Curvature Vector Fields of Submanifold. Acta Mathematica Sinica, Chinese Series, 2013, 56(5): 669-686 https://doi.org/10.12386/A2013sxxb0067

参考文献

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